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Theorem fneu 5034
Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fneu ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fneu
StepHypRef Expression
1 funmo 4947 . . . 4 (Fun 𝐹 → ∃*𝑦 𝐵𝐹𝑦)
21adantr 270 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃*𝑦 𝐵𝐹𝑦)
3 eldmg 4558 . . . . . 6 (𝐵 ∈ dom 𝐹 → (𝐵 ∈ dom 𝐹 ↔ ∃𝑦 𝐵𝐹𝑦))
43ibi 174 . . . . 5 (𝐵 ∈ dom 𝐹 → ∃𝑦 𝐵𝐹𝑦)
54adantl 271 . . . 4 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃𝑦 𝐵𝐹𝑦)
6 exmoeu2 1990 . . . 4 (∃𝑦 𝐵𝐹𝑦 → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
75, 6syl 14 . . 3 ((Fun 𝐹𝐵 ∈ dom 𝐹) → (∃*𝑦 𝐵𝐹𝑦 ↔ ∃!𝑦 𝐵𝐹𝑦))
82, 7mpbid 145 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → ∃!𝑦 𝐵𝐹𝑦)
98funfni 5030 1 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1422  wcel 1434  ∃!weu 1942  ∃*wmo 1943   class class class wbr 3793  dom cdm 4371  Fun wfun 4926   Fn wfn 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-fun 4934  df-fn 4935
This theorem is referenced by:  fneu2  5035  fnbrfvb  5246
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